Biostat/Biomath M257
System information (for reproducibility):
versioninfo()Julia Version 1.10.2
Commit bd47eca2c8a (2024-03-01 10:14 UTC)
Build Info:
Official https://julialang.org/ release
Platform Info:
OS: macOS (arm64-apple-darwin22.4.0)
CPU: 12 × Apple M2 Max
WORD_SIZE: 64
LIBM: libopenlibm
LLVM: libLLVM-15.0.7 (ORCJIT, apple-m1)
Threads: 8 default, 0 interactive, 4 GC (on 8 virtual cores)
Environment:
JULIA_NUM_THREADS = 8
JULIA_EDITOR = codeLoad packages:
using Pkg
Pkg.activate(pwd())
Pkg.instantiate()
Pkg.status() Activating project at `~/Documents/github.com/ucla-biostat-257/2024spring/slides/08-numalgintro`Status `~/Documents/github.com/ucla-biostat-257/2024spring/slides/08-numalgintro/Project.toml`
[6e4b80f9] BenchmarkTools v1.5.0
[0e44f5e4] Hwloc v3.0.1
[bdcacae8] LoopVectorization v0.12.169
[6f49c342] RCall v0.14.1
[37e2e46d] LinearAlgebra
[9a3f8284] RandomTopics in numerical algebra:
Except for the iterative methods, most of these numerical linear algebra tasks are implemented in the BLAS and LAPACK libraries. They form the building blocks of most statistical computing tasks (optimization, MCMC).
Our major goal (or learning objectives) is to
All high-level languages (Julia, Matlab, Python, R) call BLAS and LAPACK for numerical linear algebra.
BLAS stands for basic linear algebra subprograms.
See netlib for a complete list of standardized BLAS functions.
There are many implementations of BLAS.
There are 3 levels of BLAS functions.
| Level | Example Operation | Name | Dimension | Flops |
|---|---|---|---|---|
| 1 | \(\alpha \gets \mathbf{x}^T \mathbf{y}\) | dot product | \(\mathbf{x}, \mathbf{y} \in \mathbb{R}^n\) | \(2n\) |
| 1 | \(\mathbf{y} \gets \mathbf{y} + \alpha \mathbf{x}\) | axpy | \(\alpha \in \mathbb{R}\), \(\mathbf{x}, \mathbf{y} \in \mathbb{R}^n\) | \(2n\) |
| 2 | \(\mathbf{y} \gets \mathbf{y} + \mathbf{A} \mathbf{x}\) | gaxpy | \(\mathbf{A} \in \mathbb{R}^{m \times n}\), \(\mathbf{x} \in \mathbb{R}^n\), \(\mathbf{y} \in \mathbb{R}^m\) | \(2mn\) |
| 2 | \(\mathbf{A} \gets \mathbf{A} + \mathbf{y} \mathbf{x}^T\) | rank one update | \(\mathbf{A} \in \mathbb{R}^{m \times n}\), \(\mathbf{x} \in \mathbb{R}^n\), \(\mathbf{y} \in \mathbb{R}^m\) | \(2mn\) |
| 3 | \(\mathbf{C} \gets \mathbf{C} + \mathbf{A} \mathbf{B}\) | matrix multiplication | \(\mathbf{A} \in \mathbb{R}^{m \times p}\), \(\mathbf{B} \in \mathbb{R}^{p \times n}\), \(\mathbf{C} \in \mathbb{R}^{m \times n}\) | \(2mnp\) |
The form of a mathematical expression and the way the expression should be evaluated in actual practice may be quite different.
Some operations appear as level-3 but indeed are level-2.
Example 1. A common operation in statistics is column scaling or row scaling \[
\begin{eqnarray*}
\mathbf{A} &=& \mathbf{A} \mathbf{D} \quad \text{(column scaling)} \\
\mathbf{A} &=& \mathbf{D} \mathbf{A} \quad \text{(row scaling)},
\end{eqnarray*}
\] where \(\mathbf{D}\) is diagonal. For example, in generalized linear models (GLMs), the Fisher information matrix takes the form
\[
\mathbf{X}^T \mathbf{W} \mathbf{X},
\] where \(\mathbf{W}\) is a diagonal matrix with observation weights on diagonal.
Column and row scalings are essentially level-2 operations!
using BenchmarkTools, LinearAlgebra, Random
Random.seed!(257) # seed
n = 2000
A = rand(n, n) # n-by-n matrix
d = rand(n) # n vector
D = Diagonal(d) # diagonal matrix with d as diagonal2000×2000 Diagonal{Float64, Vector{Float64}}:
0.0416032 ⋅ ⋅ ⋅ … ⋅ ⋅ ⋅
⋅ 0.887679 ⋅ ⋅ ⋅ ⋅ ⋅
⋅ ⋅ 0.102233 ⋅ ⋅ ⋅ ⋅
⋅ ⋅ ⋅ 0.08407 ⋅ ⋅ ⋅
⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅
⋅ ⋅ ⋅ ⋅ … ⋅ ⋅ ⋅
⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅
⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅
⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅
⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅
⋮ ⋱
⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅
⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅
⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅
⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅
⋅ ⋅ ⋅ ⋅ … ⋅ ⋅ ⋅
⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅
⋅ ⋅ ⋅ ⋅ 0.213471 ⋅ ⋅
⋅ ⋅ ⋅ ⋅ ⋅ 0.870533 ⋅
⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 0.318876Dfull = diagm(d) # convert to full matrix2000×2000 Matrix{Float64}:
0.0416032 0.0 0.0 0.0 … 0.0 0.0 0.0
0.0 0.887679 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.102233 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.08407 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 … 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0
⋮ ⋱
0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 … 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.213471 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.870533 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.318876# this is calling BLAS routine for matrix multiplication: O(n^3) flops
# this is SLOW!
@benchmark $A * $DfullBenchmarkTools.Trial: 30 samples with 1 evaluation. Range (min … max): 110.016 ms … 330.218 ms ┊ GC (min … max): 0.00% … 0.00% Time (median): 160.076 ms ┊ GC (median): 0.00% Time (mean ± σ): 169.199 ms ± 61.926 ms ┊ GC (mean ± σ): 0.28% ± 0.57% █ ▁ █▄▄▁▁▁▁▄▄▄▄▁▇█▇▇▄▁▁▄▁▁▄▁▄▁▁▄▁▁▁▁▁▄▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▄▁▇ ▁ 110 ms Histogram: frequency by time 330 ms < Memory estimate: 30.52 MiB, allocs estimate: 2.
# dispatch to special method for diagonal matrix multiplication.
# columnwise scaling: O(n^2) flops
@benchmark $A * $DBenchmarkTools.Trial: 1786 samples with 1 evaluation. Range (min … max): 2.240 ms … 4.773 ms ┊ GC (min … max): 0.00% … 32.09% Time (median): 2.523 ms ┊ GC (median): 0.00% Time (mean ± σ): 2.797 ms ± 532.686 μs ┊ GC (mean ± σ): 10.35% ± 13.58% ▅█▃▃ ▂▃▃▅▅▇██████▆▄▄▃▃▂▂▂▂▂▂▁▁▂▁▁▂▁▂▁▁▁▁▁▁▁▁▂▂▂▂▃▃▄▅▆▆▅▄▃▃▃▃▃▂▂▂ ▃ 2.24 ms Histogram: frequency by time 3.98 ms < Memory estimate: 30.52 MiB, allocs estimate: 2.
# Or we can use broadcasting (with recycling)
@benchmark $A .* transpose($d)BenchmarkTools.Trial: 1827 samples with 1 evaluation. Range (min … max): 2.256 ms … 4.258 ms ┊ GC (min … max): 0.00% … 32.83% Time (median): 2.460 ms ┊ GC (median): 0.00% Time (mean ± σ): 2.734 ms ± 527.168 μs ┊ GC (mean ± σ): 10.63% ± 13.87% ▁█▆▆▆▂▂ ▂▃████████▇▆▄▃▃▂▃▂▂▂▂▂▂▂▁▂▁▁▂▁▁▁▁▁▁▁▁▁▁▁▂▁▁▁▂▃▄▅▅▆▆▆▅▄▃▃▃▃▃ ▃ 2.26 ms Histogram: frequency by time 3.83 ms < Memory estimate: 30.52 MiB, allocs estimate: 2.
# in-place: avoid allocate space for result
# rmul!: compute matrix-matrix product A*B, overwriting A, and return the result.
@benchmark rmul!($A, $D)BenchmarkTools.Trial: 3314 samples with 1 evaluation. Range (min … max): 1.287 ms … 4.918 ms ┊ GC (min … max): 0.00% … 0.00% Time (median): 1.504 ms ┊ GC (median): 0.00% Time (mean ± σ): 1.506 ms ± 130.657 μs ┊ GC (mean ± σ): 0.00% ± 0.00% ▄█▅ ▂▁▂▂▁▂▂▂▂▃▄▄▅▅▆▅▆▆▇████▆▄▄▃▃▃▂▂▂▂▂▂▂▂▂▂▂▂▁▂▂▁▂▂▂▂▂▂▂▂▁▁▂▂▂▂ ▃ 1.29 ms Histogram: frequency by time 1.91 ms < Memory estimate: 0 bytes, allocs estimate: 0.
# In-place broadcasting
@benchmark $A .= $A .* transpose($d)BenchmarkTools.Trial: 3051 samples with 1 evaluation. Range (min … max): 1.525 ms … 8.799 ms ┊ GC (min … max): 0.00% … 0.00% Time (median): 1.574 ms ┊ GC (median): 0.00% Time (mean ± σ): 1.636 ms ± 326.920 μs ┊ GC (mean ± σ): 0.00% ± 0.00% ▆█▇▅▄▃▂▁ ▁ █████████▇▆▇▆▆▄▆▅▅▆▃▅▆▃▅▅▆▄▅▃▁▄▄▄▅▅▄▄▅▄▃▃▅▅▃▃▄▃▄▁▃▃▁▅▁▃▁▁▅▄ █ 1.53 ms Histogram: log(frequency) by time 2.99 ms < Memory estimate: 0 bytes, allocs estimate: 0.
Exercise: Try @turbo (SIMD) and @tturbo (SIMD) from LoopVectorization.jl package.
Note: In R or Matlab, diag(d) will create a full matrix. Be cautious using diag function: do we really need a full diagonal matrix?
using RCall
R"""
d <- runif(5)
diag(d)
"""RObject{RealSxp}
[,1] [,2] [,3] [,4] [,5]
[1,] 0.9431375 0.0000000 0.0000000 0.00000000 0.0000000
[2,] 0.0000000 0.1546795 0.0000000 0.00000000 0.0000000
[3,] 0.0000000 0.0000000 0.3322724 0.00000000 0.0000000
[4,] 0.0000000 0.0000000 0.0000000 0.08111416 0.0000000
[5,] 0.0000000 0.0000000 0.0000000 0.00000000 0.9322544# This works only when Matlab is installed
using MATLAB
mat"""
d = rand(5, 1)
diag(d)
"""ArgumentError: ArgumentError: Package MATLAB not found in current path.
- Run `import Pkg; Pkg.add("MATLAB")` to install the MATLAB package.Example 2. Innter product between two matrices \(\mathbf{A}, \mathbf{B} \in \mathbb{R}^{m \times n}\) is often written as \[ \text{trace}(\mathbf{A}^T \mathbf{B}), \text{trace}(\mathbf{B} \mathbf{A}^T), \text{trace}(\mathbf{A} \mathbf{B}^T), \text{ or } \text{trace}(\mathbf{B}^T \mathbf{A}). \] They appear as level-3 operation (matrix multiplication with \(O(m^2n)\) or \(O(mn^2)\) flops).
Random.seed!(123)
n = 2000
A, B = randn(n, n), randn(n, n)
# slow way to evaluate tr(A'B): 2mn^2 flops
@benchmark tr(transpose($A) * $B)BenchmarkTools.Trial: 62 samples with 1 evaluation. Range (min … max): 50.385 ms … 147.322 ms ┊ GC (min … max): 0.00% … 0.00% Time (median): 66.011 ms ┊ GC (median): 0.00% Time (mean ± σ): 81.664 ms ± 31.991 ms ┊ GC (mean ± σ): 0.58% ± 1.40% █ ▁ █▅█▁▅▇▁▁▁▅▁▅▇▁▅▁▁▅▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▅▁▁▁▅▅▅▁▅▅▁▁▇▅▇▇▇▇▇▇▁▇▁▇▁▅ ▁ 50.4 ms Histogram: log(frequency) by time 130 ms < Memory estimate: 30.52 MiB, allocs estimate: 2.
But \(\text{trace}(\mathbf{A}^T \mathbf{B}) = <\text{vec}(\mathbf{A}), \text{vec}(\mathbf{B})>\). The latter is level-1 BLAS operation with \(O(mn)\) flops.
# smarter way to evaluate tr(A'B): 2mn flops
@benchmark dot($A, $B)BenchmarkTools.Trial: 2698 samples with 1 evaluation. Range (min … max): 1.793 ms … 2.206 ms ┊ GC (min … max): 0.00% … 0.00% Time (median): 1.836 ms ┊ GC (median): 0.00% Time (mean ± σ): 1.850 ms ± 48.023 μs ┊ GC (mean ± σ): 0.00% ± 0.00% ▂▅███▇▇▇▆▆▆▅▅▅▃▃▂▂ ▂ ▃▃▆█████████████████████▇▆▆▆▇▅▅▆▅▆▅▅▅▇▅▅▆▆▅▆▅▆▆█▆▆▆▅▆▇▅▆▄▆ █ 1.79 ms Histogram: log(frequency) by time 2.05 ms < Memory estimate: 0 bytes, allocs estimate: 0.
Example 3. Similarly \(\text{diag}(\mathbf{A}^T \mathbf{B})\) can be calculated in \(O(mn)\) flops.
# slow way to evaluate diag(A'B): O(n^3)
@benchmark diag(transpose($A) * $B)BenchmarkTools.Trial: 93 samples with 1 evaluation. Range (min … max): 50.588 ms … 75.902 ms ┊ GC (min … max): 0.00% … 0.00% Time (median): 52.324 ms ┊ GC (median): 0.00% Time (mean ± σ): 53.967 ms ± 4.768 ms ┊ GC (mean ± σ): 0.89% ± 1.54% █ ▁ ▂ █▇█▇▆█▃▃▄▃▃▁▁▃▃▃▃▄▃▃▃▁▃▃▁▁▁▁▁▃▁▁▃▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▃ ▁ 50.6 ms Histogram: frequency by time 74 ms < Memory estimate: 30.53 MiB, allocs estimate: 3.
# smarter way to evaluate diag(A'B): O(n^2)
@benchmark Diagonal(vec(sum($A .* $B, dims = 1)))BenchmarkTools.Trial: 1257 samples with 1 evaluation. Range (min … max): 3.330 ms … 15.680 ms ┊ GC (min … max): 0.00% … 9.60% Time (median): 3.640 ms ┊ GC (median): 0.00% Time (mean ± σ): 3.972 ms ± 774.977 μs ┊ GC (mean ± σ): 9.38% ± 12.67% ▄█▅▃▁ ▅███████▇▇▅▅▄▃▃▃▂▂▂▂▂▂▁▂▂▁▁▂▁▂▃▅▆▃▄▄▄▄▄▃▃▂▃▃▂▂▂▂▂▂▂▂▂▃▃▃▃▂▂ ▃ 3.33 ms Histogram: frequency by time 5.93 ms < Memory estimate: 30.53 MiB, allocs estimate: 5.
To get rid of allocation of intermediate arrays at all, we can just write a double loop or use dot function.
function diag_matmul!(d, A, B)
m, n = size(A)
@assert size(B) == (m, n) "A and B should have same size"
fill!(d, 0)
for j in 1:n, i in 1:m
d[j] += A[i, j] * B[i, j]
end
Diagonal(d)
end
d = zeros(eltype(A), size(A, 2))
@benchmark diag_matmul!($d, $A, $B)BenchmarkTools.Trial: 1465 samples with 1 evaluation. Range (min … max): 3.319 ms … 13.782 ms ┊ GC (min … max): 0.00% … 0.00% Time (median): 3.374 ms ┊ GC (median): 0.00% Time (mean ± σ): 3.409 ms ± 509.119 μs ┊ GC (mean ± σ): 0.00% ± 0.00% ▅▄ ▁▃▂▃▇▇▇▇██▆▃▆▄▄ ▁▂ ██▇███████████████▇██▆▆▆▄▅▅▅▅▅▄▃▃▃▃▃▃▂▃▃▂▁▂▂▁▁▂▁▁▁▁▂▂▁▁▂▁▂▂ ▄ 3.32 ms Histogram: frequency by time 3.58 ms < Memory estimate: 0 bytes, allocs estimate: 0.
Exercise: Try @turbo (SIMD) and @tturbo (SIMD) from LoopVectorization.jl package.
Key to high performance is effective use of memory hierarchy. True on all architectures.
Source: https://cs.brown.edu/courses/csci1310/2020/assign/labs/lab4.html
Hwloc.jl package. For example, this laptop runs an Apple M2 Max chip with 4 efficiency cores and 8 performance cores.using Hwloc
topology_graphical()/------------------------------------------------------------------------------------------------------------------------------------------------------------------\
| Machine (2208MB total) |
| |
| /----------------------------------------------------------------------------------------------------------------------------------------\ /------------------\ |
| | Package L#0 | | CoProc opencl0d0 | |
| | | | | |
| | /------------------------------------------------------------------------------------------------------------------------------------\ | | 38 compute units | |
| | | NUMANode L#0 P#0 (2208MB) | | | | |
| | \------------------------------------------------------------------------------------------------------------------------------------/ | | 72 GB | |
| | | \------------------/ |
| | /----------------------------------------------------------------\ /----------------------------------------------------------------\ | |
| | | L2 (4096KB) | | L2 (16MB) | | |
| | \----------------------------------------------------------------/ \----------------------------------------------------------------/ | |
| | | |
| | /-------------\ /-------------\ /-------------\ /-------------\ /-------------\ /-------------\ /-------------\ /-------------\ | |
| | | L1d (64KB) | | L1d (64KB) | | L1d (64KB) | | L1d (64KB) | | L1d (128KB) | | L1d (128KB) | | L1d (128KB) | | L1d (128KB) | | |
| | \-------------/ \-------------/ \-------------/ \-------------/ \-------------/ \-------------/ \-------------/ \-------------/ | |
| | | |
| | /-------------\ /-------------\ /-------------\ /-------------\ /-------------\ /-------------\ /-------------\ /-------------\ | |
| | | L1i (128KB) | | L1i (128KB) | | L1i (128KB) | | L1i (128KB) | | L1i (192KB) | | L1i (192KB) | | L1i (192KB) | | L1i (192KB) | | |
| | \-------------/ \-------------/ \-------------/ \-------------/ \-------------/ \-------------/ \-------------/ \-------------/ | |
| | | |
| | /-------------\ /-------------\ /-------------\ /-------------\ /-------------\ /-------------\ /-------------\ /-------------\ | |
| | | Core L#0 | | Core L#1 | | Core L#2 | | Core L#3 | | Core L#4 | | Core L#5 | | Core L#6 | | Core L#7 | | |
| | | | | | | | | | | | | | | | | | | |
| | | /---------\ | | /---------\ | | /---------\ | | /---------\ | | /---------\ | | /---------\ | | /---------\ | | /---------\ | | |
| | | | PU L#0 | | | | PU L#1 | | | | PU L#2 | | | | PU L#3 | | | | PU L#4 | | | | PU L#5 | | | | PU L#6 | | | | PU L#7 | | | |
| | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | |
| | | | P#0 | | | | P#1 | | | | P#2 | | | | P#3 | | | | P#4 | | | | P#5 | | | | P#6 | | | | P#7 | | | |
| | | \---------/ | | \---------/ | | \---------/ | | \---------/ | | \---------/ | | \---------/ | | \---------/ | | \---------/ | | |
| | \-------------/ \-------------/ \-------------/ \-------------/ \-------------/ \-------------/ \-------------/ \-------------/ | |
| | | |
| | /----------------------------------------------------------------\ | |
| | | L2 (16MB) | | |
| | \----------------------------------------------------------------/ | |
| | | |
| | /-------------\ /-------------\ /-------------\ /-------------\ | |
| | | L1d (128KB) | | L1d (128KB) | | L1d (128KB) | | L1d (128KB) | | |
| | \-------------/ \-------------/ \-------------/ \-------------/ | |
| | | |
| | /-------------\ /-------------\ /-------------\ /-------------\ | |
| | | L1i (192KB) | | L1i (192KB) | | L1i (192KB) | | L1i (192KB) | | |
| | \-------------/ \-------------/ \-------------/ \-------------/ | |
| | | |
| | /-------------\ /-------------\ /-------------\ /-------------\ | |
| | | Core L#8 | | Core L#9 | | Core L#10 | | Core L#11 | | |
| | | | | | | | | | | |
| | | /---------\ | | /---------\ | | /---------\ | | /---------\ | | |
| | | | PU L#8 | | | | PU L#9 | | | | PU L#10 | | | | PU L#11 | | | |
| | | | | | | | | | | | | | | | | | | |
| | | | P#8 | | | | P#9 | | | | P#10 | | | | P#11 | | | |
| | | \---------/ | | \---------/ | | \---------/ | | \---------/ | | |
| | \-------------/ \-------------/ \-------------/ \-------------/ | |
| \----------------------------------------------------------------------------------------------------------------------------------------/ |
\------------------------------------------------------------------------------------------------------------------------------------------------------------------/For example, Xeon X5650 CPU has a theoretical throughput of 128 DP GFLOPS but a max memory bandwidth of 32GB/s.
Can we keep CPU cores busy with enough deliveries of matrix data and ship the results to memory fast enough to avoid backlog?
Answer: use high-level BLAS as much as possible.
| BLAS | Dimension | Mem. Refs. | Flops | Ratio |
|---|---|---|---|---|
| Level 1: \(\mathbf{y} \gets \mathbf{y} + \alpha \mathbf{x}\) | \(\mathbf{x}, \mathbf{y} \in \mathbb{R}^n\) | \(3n\) | \(2n\) | 3:2 |
| Level 2: \(\mathbf{y} \gets \mathbf{y} + \mathbf{A} \mathbf{x}\) | \(\mathbf{x}, \mathbf{y} \in \mathbb{R}^n\), \(\mathbf{A} \in \mathbb{R}^{n \times n}\) | \(n^2\) | \(2n^2\) | 1:2 |
| Level 3: \(\mathbf{C} \gets \mathbf{C} + \mathbf{A} \mathbf{B}\) | \(\mathbf{A}, \mathbf{B}, \mathbf{C} \in\mathbb{R}^{n \times n}\) | \(4n^2\) | \(2n^3\) | 2:n |
Source: Jack Dongarra’s slides.
A distinction between LAPACK and LINPACK (older version of R uses LINPACK) is that LAPACK makes use of higher level BLAS as much as possible (usually by smart partitioning) to increase the so-called level-3 fraction.
To appreciate the efforts in an optimized BLAS implementation such as OpenBLAS (evolved from GotoBLAS), see the Quora question, especially the video. Bottomline is
Get familiar with (good implementations of) BLAS/LAPACK and use them as much as possible.
Data layout in memory affects algorithmic efficiency too. It is much faster to move chunks of data in memory than retrieving/writing scattered data.
Storage mode: column-major (Fortran, Matlab, R, Julia) vs row-major (C/C++).
Cache line is the minimum amount of cache which can be loaded and stored to memory.
# Apple Silicon (M1/M2 chips) don't have L3 cache
Hwloc.cachelinesize()ErrorException: Your system doesn't seem to have an L3 cache.Accessing column-major stored matrix by rows (\(ij\) looping) causes lots of cache misses.
Take matrix multiplication as an example \[ \mathbf{C} \gets \mathbf{C} + \mathbf{A} \mathbf{B}, \quad \mathbf{A} \in \mathbb{R}^{m \times p}, \mathbf{B} \in \mathbb{R}^{p \times n}, \mathbf{C} \in \mathbb{R}^{m \times n}. \] Assume the storage is column-major, such as in Julia. There are 6 variants of the algorithms according to the order in the triple loops.
jki or kji looping:# inner most loop
for i in 1:m
C[i, j] = C[i, j] + A[i, k] * B[k, j]
end- `ikj` or `kij` looping:# inner most loop
for j in 1:n
C[i, j] = C[i, j] + A[i, k] * B[k, j]
endijk or jik looping:# inner most loop
for k in 1:p
C[i, j] = C[i, j] + A[i, k] * B[k, j]
end| Variant | A Stride | B Stride | C Stride |
|---|---|---|---|
| \(jki\) or \(kji\) | Unit | 0 | Unit |
| \(ikj\) or \(kij\) | 0 | Non-Unit | Non-Unit |
| \(ijk\) or \(jik\) | Non-Unit | Unit | 0 |
Apparently the variants \(jki\) or \(kji\) are preferred.
"""
matmul_by_loop!(A, B, C, order)
Overwrite `C` by `A * B`. `order` indicates the looping order for triple loop.
"""
function matmul_by_loop!(A::Matrix, B::Matrix, C::Matrix, order::String)
m = size(A, 1)
p = size(A, 2)
n = size(B, 2)
fill!(C, 0)
if order == "jki"
@inbounds for j = 1:n, k = 1:p, i = 1:m
C[i, j] += A[i, k] * B[k, j]
end
end
if order == "kji"
@inbounds for k = 1:p, j = 1:n, i = 1:m
C[i, j] += A[i, k] * B[k, j]
end
end
if order == "ikj"
@inbounds for i = 1:m, k = 1:p, j = 1:n
C[i, j] += A[i, k] * B[k, j]
end
end
if order == "kij"
@inbounds for k = 1:p, i = 1:m, j = 1:n
C[i, j] += A[i, k] * B[k, j]
end
end
if order == "ijk"
@inbounds for i = 1:m, j = 1:n, k = 1:p
C[i, j] += A[i, k] * B[k, j]
end
end
if order == "jik"
@inbounds for j = 1:n, i = 1:m, k = 1:p
C[i, j] += A[i, k] * B[k, j]
end
end
end
using Random
Random.seed!(123)
m, p, n = 2000, 100, 2000
A = rand(m, p)
B = rand(p, n)
C = zeros(m, n);using BenchmarkTools
@benchmark matmul_by_loop!($A, $B, $C, "jki")BenchmarkTools.Trial: 86 samples with 1 evaluation. Range (min … max): 57.729 ms … 70.826 ms ┊ GC (min … max): 0.00% … 0.00% Time (median): 58.222 ms ┊ GC (median): 0.00% Time (mean ± σ): 58.433 ms ± 1.412 ms ┊ GC (mean ± σ): 0.00% ± 0.00% ▁ █ ▆ ▁▁▁▁ ▁ ▃ ▁ ▆ ▁ ▇█▇▁█▇▄█▇▇████▄▇▁█▇▄▇█▄▇▄▄▁▇█▁▁█▄▄▁▁█▄▄▄▄▁▇▁▁▁▁▁▄▄▁▄▁▁▁▇▄▁▄ ▁ 57.7 ms Histogram: frequency by time 59.3 ms < Memory estimate: 0 bytes, allocs estimate: 0.
@benchmark matmul_by_loop!($A, $B, $C, "kji")BenchmarkTools.Trial: 27 samples with 1 evaluation. Range (min … max): 183.530 ms … 212.516 ms ┊ GC (min … max): 0.00% … 0.00% Time (median): 184.559 ms ┊ GC (median): 0.00% Time (mean ± σ): 186.442 ms ± 5.699 ms ┊ GC (mean ± σ): 0.00% ± 0.00% █▆▄ ███▁▁▆▄▆▁▁▁▄▁▁▁▄▁▁▁▄▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▄ ▁ 184 ms Histogram: frequency by time 213 ms < Memory estimate: 0 bytes, allocs estimate: 0.
@benchmark matmul_by_loop!($A, $B, $C, "ikj")BenchmarkTools.Trial: 10 samples with 1 evaluation. Range (min … max): 509.252 ms … 527.728 ms ┊ GC (min … max): 0.00% … 0.00% Time (median): 515.239 ms ┊ GC (median): 0.00% Time (mean ± σ): 515.454 ms ± 5.554 ms ┊ GC (mean ± σ): 0.00% ± 0.00% █ ▁ ▁ ▁ ▁ ▁ ▁▁ ▁ █▁▁▁▁▁█▁▁▁█▁▁▁▁▁█▁▁▁▁▁█▁▁█▁▁▁▁██▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁█ ▁ 509 ms Histogram: frequency by time 528 ms < Memory estimate: 0 bytes, allocs estimate: 0.
@benchmark matmul_by_loop!($A, $B, $C, "kij")BenchmarkTools.Trial: 10 samples with 1 evaluation. Range (min … max): 507.229 ms … 530.723 ms ┊ GC (min … max): 0.00% … 0.00% Time (median): 511.375 ms ┊ GC (median): 0.00% Time (mean ± σ): 513.003 ms ± 6.944 ms ┊ GC (mean ± σ): 0.00% ± 0.00% ████ █ █ █ ██ █ ████▁▁▁▁▁█▁█▁▁▁▁▁█▁▁██▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁█ ▁ 507 ms Histogram: frequency by time 531 ms < Memory estimate: 0 bytes, allocs estimate: 0.
@benchmark matmul_by_loop!($A, $B, $C, "ijk")BenchmarkTools.Trial: 21 samples with 1 evaluation. Range (min … max): 244.667 ms … 265.676 ms ┊ GC (min … max): 0.00% … 0.00% Time (median): 247.189 ms ┊ GC (median): 0.00% Time (mean ± σ): 249.187 ms ± 5.163 ms ┊ GC (mean ± σ): 0.00% ± 0.00% █▃▃ ▃ ▃ ▇███▁▁█▇▇█▁▁▁▁▁▇▁▁▁▁▁▁▇▁▇▁▇▁▇▁▁▁▁▇▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▇ ▁ 245 ms Histogram: frequency by time 266 ms < Memory estimate: 0 bytes, allocs estimate: 0.
@benchmark matmul_by_loop!($A, $B, $C, "ijk")BenchmarkTools.Trial: 21 samples with 1 evaluation. Range (min … max): 245.461 ms … 255.071 ms ┊ GC (min … max): 0.00% … 0.00% Time (median): 247.440 ms ┊ GC (median): 0.00% Time (mean ± σ): 248.889 ms ± 3.108 ms ┊ GC (mean ± σ): 0.00% ± 0.00% █▁▁ ▁▁▁▁█ ▁▁ ▁ ▁ ▁ ▁ ▁ ▁ ▁▁ ▁ ███▁▁▁█████▁██▁▁▁▁▁▁▁▁▁█▁▁█▁▁▁▁█▁▁▁▁▁█▁▁▁▁▁▁▁█▁█▁▁██▁▁▁▁▁▁▁▁█ ▁ 245 ms Histogram: frequency by time 255 ms < Memory estimate: 0 bytes, allocs estimate: 0.
@benchmark mul!($C, $A, $B)BenchmarkTools.Trial: 1708 samples with 1 evaluation. Range (min … max): 2.526 ms … 24.014 ms ┊ GC (min … max): 0.00% … 0.00% Time (median): 2.643 ms ┊ GC (median): 0.00% Time (mean ± σ): 2.926 ms ± 1.033 ms ┊ GC (mean ± σ): 0.00% ± 0.00% ██▆▂▃▂▂▃▃▁▁▁ ▁ ▁ ████████████████▇▇█▇▇▇▆▆▅▆▄▄▆▆▁▁▆▄▄▅▁▁▄▄▄▁▁▄▄▁▁▁▄▁▅▁▁▁▄▄▄▄ █ 2.53 ms Histogram: log(frequency) by time 6.75 ms < Memory estimate: 0 bytes, allocs estimate: 0.
# direct call of BLAS wrapper function
@benchmark LinearAlgebra.BLAS.gemm!('N', 'N', 1.0, $A, $B, 0.0, $C)BenchmarkTools.Trial: 1653 samples with 1 evaluation. Range (min … max): 2.554 ms … 26.166 ms ┊ GC (min … max): 0.00% … 0.00% Time (median): 2.672 ms ┊ GC (median): 0.00% Time (mean ± σ): 3.022 ms ± 1.278 ms ┊ GC (mean ± σ): 0.00% ± 0.00% ██▄▂▂▃▂▂▂▂▁ ▁ ████████████████▅▇▆▆▆▆▅▇▆▇▄▄▄▅▄▄▁▄▄▄▄▄▁▄▄▄▄▄▁▄▁▄▁▄▄▄▁▄▄▄▁▄ █ 2.55 ms Histogram: log(frequency) by time 7.47 ms < Memory estimate: 0 bytes, allocs estimate: 0.
Question (again): Can our loop beat BLAS?
Exercise: Annotate the loop in matmul_by_loop! by @turbo and @tturbo (multi-threading) and benchmark again.
using RCall
R"""
sessionInfo()
"""RObject{VecSxp}
R version 4.3.2 (2023-10-31)
Platform: aarch64-apple-darwin20 (64-bit)
Running under: macOS Sonoma 14.4.1
Matrix products: default
BLAS: /System/Library/Frameworks/Accelerate.framework/Versions/A/Frameworks/vecLib.framework/Versions/A/libBLAS.dylib
LAPACK: /Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/lib/libRlapack.dylib; LAPACK version 3.11.0
locale:
[1] C
time zone: America/Los_Angeles
tzcode source: internal
attached base packages:
[1] stats graphics grDevices utils datasets methods base
loaded via a namespace (and not attached):
[1] compiler_4.3.2R"""
library(dplyr)
library(bench)
A <- $A
B <- $B
bench::mark(A %*% B) %>%
print(width = Inf)
""";┌ Warning: RCall.jl:
│ Attaching package: 'dplyr'
│
│ The following objects are masked from 'package:stats':
│
│ filter, lag
│
│ The following objects are masked from 'package:base':
│
│ intersect, setdiff, setequal, union
│
└ @ RCall /Users/huazhou/.julia/packages/RCall/dDAVd/src/io.jl:172
┌ Warning: RCall.jl: Warning: Some expressions had a GC in every iteration; so filtering is disabled.
└ @ RCall /Users/huazhou/.julia/packages/RCall/dDAVd/src/io.jl:172# A tibble: 1 x 13
expression min median `itr/sec` mem_alloc `gc/sec` n_itr n_gc
<bch:expr> <bch:tm> <bch:tm> <dbl> <bch:byt> <dbl> <int> <dbl>
1 A %*% B 124ms 125ms 7.75 30.5MB 7.75 4 4
total_time result memory time
<bch:tm> <list> <list> <list>
1 516ms <dbl [2,000 x 2,000]> <Rprofmem [1 x 3]> <bench_tm [4]>
gc
<list>
1 <tibble [4 x 3]>Re-build R from source using OpenBLAS or MKL will immediately boost linear algebra performance in R. Google build R using MKL to get started. Similarly we can build Julia using MKL.
Matlab uses MKL. Usually it’s very hard to beat Matlab in terms of linear algebra.
using MATLAB
mat"""
f = @() $A * $B;
timeit(f)
"""ArgumentError: ArgumentError: Package MATLAB not found in current path.
- Run `import Pkg; Pkg.add("MATLAB")` to install the MATLAB package.In R, the command
t(A) %*% xwill first transpose A then perform matrix multiplication, causing unnecessary memory allocation
using Random, LinearAlgebra, BenchmarkTools
Random.seed!(123)
n = 1000
A = rand(n, n)
x = rand(n);R"""
A <- $A
x <- $x
bench::mark(t(A) %*% x) %>%
print(width = Inf)
""";# A tibble: 1 x 13
expression min median `itr/sec` mem_alloc `gc/sec` n_itr n_gc
<bch:expr> <bch:tm> <bch:tm> <dbl> <bch:byt> <dbl> <int> <dbl>
1 t(A) %*% x 2.1ms 2.22ms 446. 7.64MB 91.4 166 34
total_time result memory time
<bch:tm> <list> <list> <list>
1 372ms <dbl [1,000 x 1]> <Rprofmem [2 x 3]> <bench_tm [200]>
gc
<list>
1 <tibble [200 x 3]>Julia is avoids transposing matrix whenever possible. The Transpose type only creates a view of the transpose of matrix data.
typeof(transpose(A))Transpose{Float64, Matrix{Float64}}fieldnames(typeof(transpose(A)))(:parent,)# same data in tranpose(A) and original matrix A
pointer(transpose(A).parent), pointer(A)(Ptr{Float64} @0x0000000129840000, Ptr{Float64} @0x0000000129840000)# dispatch to BLAS
# does *not* actually transpose the matrix
@benchmark transpose($A) * $xBenchmarkTools.Trial: 10000 samples with 1 evaluation. Range (min … max): 22.625 μs … 1.402 ms ┊ GC (min … max): 0.00% … 0.00% Time (median): 22.959 μs ┊ GC (median): 0.00% Time (mean ± σ): 25.961 μs ± 24.431 μs ┊ GC (mean ± σ): 0.00% ± 0.00% █▇▃ ▁ ▁ ████▇▅▅▆▅▅▆▇▇▇▇▇▆▆▆▆▆▆▆▆▅▅▆▅▆▅▆▅▅▅▁▅███▇▅▅▆▆▆▅▅▆▅▆▅▅▅▄▄▄▃▄▅ █ 22.6 μs Histogram: log(frequency) by time 65.8 μs < Memory estimate: 8.00 KiB, allocs estimate: 1.
# pre-allocate result
out = zeros(size(A, 2))
@benchmark mul!($out, transpose($A), $x)BenchmarkTools.Trial: 10000 samples with 1 evaluation. Range (min … max): 22.542 μs … 2.993 ms ┊ GC (min … max): 0.00% … 0.00% Time (median): 22.834 μs ┊ GC (median): 0.00% Time (mean ± σ): 26.407 μs ± 54.119 μs ┊ GC (mean ± σ): 0.00% ± 0.00% █▂▁ ▁ ▁ ████▆██▇▆▇▇▆▇▇▇▇▆▆▅▄▄▅▄▅▄▄▄▅▅▅▄▅▄▄▄▄▁▄▁▄▅▇▄▆▆▇▇███▆▄▄▃▄▄▄▅▅ █ 22.5 μs Histogram: log(frequency) by time 56.7 μs < Memory estimate: 0 bytes, allocs estimate: 0.
# or call BLAS wrapper directly
@benchmark LinearAlgebra.BLAS.gemv!('T', 1.0, $A, $x, 0.0, $out)BenchmarkTools.Trial: 10000 samples with 1 evaluation. Range (min … max): 22.666 μs … 86.333 μs ┊ GC (min … max): 0.00% … 0.00% Time (median): 22.875 μs ┊ GC (median): 0.00% Time (mean ± σ): 23.589 μs ± 2.546 μs ┊ GC (mean ± σ): 0.00% ± 0.00% █▅▃▃▅▄▂▁ ▁ ████████▆▆▅▄▁██▆█▇███▆▇▇▇█▆▇▅▅▅▆▆▆▅▆▅▅▆▅▆▆▆▆▇▅▅▅▄▄▁▃▄▄▄▅▄▅▅ █ 22.7 μs Histogram: log(frequency) by time 35.2 μs < Memory estimate: 0 bytes, allocs estimate: 0.
Broadcasting in Julia achieves vectorized code without creating intermediate arrays.
Suppose we want to calculate elementsize maximum of absolute values of two large arrays. In R or Matlab, the command
max(abs(X), abs(Y))will create two intermediate arrays and then one result array.
using RCall
Random.seed!(123)
X, Y = rand(1000, 1000), rand(1000, 1000)
R"""
library(dplyr)
library(bench)
bench::mark(max(abs($X), abs($Y))) %>%
print(width = Inf)
""";# A tibble: 1 x 13
expression min median `itr/sec` mem_alloc `gc/sec`
<bch:expr> <bch:tm> <bch:tm> <dbl> <bch:byt> <dbl>
1 max(abs(`#JL`$X), abs(`#JL`$Y)) 2.85ms 3.74ms 269. 15.3MB 227.
n_itr n_gc total_time result memory time
<int> <dbl> <bch:tm> <list> <list> <list>
1 63 53 234ms <dbl [1]> <Rprofmem [2 x 3]> <bench_tm [116]>
gc
<list>
1 <tibble [116 x 3]>In Julia, dot operations are fused so no intermediate arrays are created.
# no intermediate arrays created, only result array created
@benchmark max.(abs.($X), abs.($Y))BenchmarkTools.Trial: 5994 samples with 1 evaluation. Range (min … max): 230.583 μs … 6.933 ms ┊ GC (min … max): 0.00% … 55.37% Time (median): 736.792 μs ┊ GC (median): 0.00% Time (mean ± σ): 832.411 μs ± 522.295 μs ┊ GC (mean ± σ): 15.75% ± 18.35% ▅▅▂ ▆██▇▅▄▃▂ ▁ ▁ ▂ ███▇▆▁▃▁▅█████████▆▆▃▄▄▆▇▆▆▆▅▆▇▆▆▆▅▅▅▅▅▆████▇▇██▇▇███▇█▆▆▆▆▄▅ █ 231 μs Histogram: log(frequency) by time 2.92 ms < Memory estimate: 7.63 MiB, allocs estimate: 2.
Pre-allocating result array gets rid of memory allocation at all.
# no memory allocation at all!
Z = zeros(size(X)) # zero matrix of same size as X
@benchmark $Z .= max.(abs.($X), abs.($Y)) # .= (vs =) is important!BenchmarkTools.Trial: 10000 samples with 1 evaluation. Range (min … max): 213.292 μs … 573.292 μs ┊ GC (min … max): 0.00% … 0.00% Time (median): 229.792 μs ┊ GC (median): 0.00% Time (mean ± σ): 233.820 μs ± 11.851 μs ┊ GC (mean ± σ): 0.00% ± 0.00% ▃█▇▂ ▁▃▅▄▃▂▃▃▂▁▂▂▂▂▃▆████▇▇▆▄▃▂▂▂▂▂▂▂▂▂▂▂▄█▇▄▄▅▄▃▂▂▂▂▂▂▂▁▂▁▁▁▁▁▁▁▁ ▂ 213 μs Histogram: frequency by time 264 μs < Memory estimate: 0 bytes, allocs estimate: 0.
Exercise: Annotate the broadcasting by @turbo and @tturbo (multi-threading) and benchmark again.
View avoids creating extra copy of matrix data.
Random.seed!(123)
A = randn(1000, 1000)
# sum entries in a sub-matrix
@benchmark sum($A[1:2:500, 1:2:500])BenchmarkTools.Trial: 10000 samples with 1 evaluation. Range (min … max): 45.291 μs … 4.860 ms ┊ GC (min … max): 0.00% … 97.70% Time (median): 68.750 μs ┊ GC (median): 0.00% Time (mean ± σ): 74.404 μs ± 121.537 μs ┊ GC (mean ± σ): 10.36% ± 6.73% █ ▁ ▆▃ █▅▁▁▁▂█▄▁▁▂▂▁▁▁▁▁▁▁▁▁▁▁▃██▅▅▆▆▇▆▇▆▆▅▅▅▅▄▄▄▄▃▃▂▂▂▂▂▂▂▂▂▁▁▁▁▁▁ ▂ 45.3 μs Histogram: frequency by time 93.4 μs < Memory estimate: 488.42 KiB, allocs estimate: 2.
# view avoids creating a separate sub-matrix
# unfortunately it's much slower possibly because of cache misses
@benchmark sum(@view $A[1:2:500, 1:2:500])BenchmarkTools.Trial: 10000 samples with 1 evaluation. Range (min … max): 54.958 μs … 207.333 μs ┊ GC (min … max): 0.00% … 0.00% Time (median): 55.167 μs ┊ GC (median): 0.00% Time (mean ± σ): 56.099 μs ± 3.432 μs ┊ GC (mean ± σ): 0.00% ± 0.00% █▇▅▃▃▅▃▁▁ ▁▁ ▁ ███████████▇███████████▇▇▇▇▇▇▇▇▇▆▇▇█▇▇▆▆▆▆▆▇▆▅▆▆▅▅▅▅▅▆▅▃▄▅▅▅ █ 55 μs Histogram: log(frequency) by time 65.8 μs < Memory estimate: 0 bytes, allocs estimate: 0.
The @views macro, which can be useful in some operations.
@benchmark @views sum($A[1:2:500, 1:2:500])BenchmarkTools.Trial: 10000 samples with 1 evaluation. Range (min … max): 54.958 μs … 4.078 ms ┊ GC (min … max): 0.00% … 0.00% Time (median): 55.250 μs ┊ GC (median): 0.00% Time (mean ± σ): 58.562 μs ± 69.852 μs ┊ GC (mean ± σ): 0.00% ± 0.00% █▇▅▃▄▂ ▁▁▁ ▂▂▁ ▁ ▁ ██████▇▆▇▆███▇███████▇▆▇▆▆▆▇▆▆▆▇▇▇▇▆▆▅▆▇▅▆▆▅▄▄▅▄▄▅▅▅▃▅▅▄▄▃▅ █ 55 μs Histogram: log(frequency) by time 69 μs < Memory estimate: 0 bytes, allocs estimate: 0.
Exercise: Here we saw that, although we avoids memory allocation, the actual run times are longer. Why?